# About Thom theorem (representation submanifold for $H_{n-2}(M)$)

Recall Thom theorem : If $M^n$ is a smooth orientable closed manifold then any homology class in $H_{n-2}(M)$ is represented by the fundamental class of a smooth submanifold.

And in the Harper and Greenberg's book there exists a following statement :

The representation of $H_2(M^4)$ be embedded surfaces is an important open problem, wheren $M^4$ is a closed $4$-dimensional manifold.

(1) Here I want to know anything related with them.

(2) If $S$ is an embedded surface in $M^4$, then is it important to know the exact value $[S]$ in $H_2(M^4)$ ?

(3) If finding a surface representing $n$ in $H_2(M^4)$ is a problem, then what is the known result ?

(4) In the paper four-manifolds which admit ${\bf Z}_p \times {\bf Z}_p$ actions - McCooey (See Section 2 in page 2), he state Edmonds's result :

If ${\bf Z}_p \times {\bf Z}_p$ acts locally linearly, homologically trivially on a closed $M^4$ with $H_1(M)=0$ and $b_2(M)\geq 1$, then each nonzero element has a fixed point set consisting of isolated points and $2$-spheres, and each sphere represents a nontrivial homology class.

Here the fixed surface $S^2$ represents $1$ or $-1$ ?

-
Where is this in Harper and Greenberg? –  Max Nov 18 '12 at 0:34
Line 12 - 13th line from the bottom of 68 page in Section 12 singular homology theory in Greenberg and Harper's book, Algebraic Topology. These questions are not in the book. Just two lines. –  Hee Kwon Lee Nov 18 '12 at 1:59